Inventory model with stock-level dependent demand rate and quantity based holding cost

Inventory model with stock-level dependent demand rate and quantity based holding cost

P. H. Bhathavala2and K. D. Rathod1

1Assistant Professor, Department of Mathematics, Sankalchand PatelCollege of Engineering, Visnagar-384315, Gujarat State, India.

2Prof. & Head, Department of Mathematics, BIT, Vadodara, Gujarat State, India.

Abstract

Inventory models in which the demand rate depends on the inventory level are based on the common real-life observation that greater product availability tends to stimulate more sales. Previous models incorporating stock-level dependent demand rate assume that the holding cost is constant for the entire inventory cycle. In this model we will discuss a stock-level dependent demand rate and a storage-time dependent holding cost. The holding cost per unit of the item per unit time is assumed to bedecreasing function of the quantity in storage. Procedures are developed for determining the optimal order quantity and the optimal cycle time.

Keywords: Inventory models, Stock-dependent demand,quantity dependent holding cost, Optimization

1. Introduction

   In traditional inventory models, the demand rate is assumed to be a given constant. Various inventory models have been developed for dealing with varying and stochastic demand. All these models implicitly assume that the demand rate is independent, i.e. an external parameter not influenced by the internal inventory policy. In real life, however, it is frequently observed that demand for a particular product can indeed be influenced by internal factors such as price and availability. The change in the demand in response to inventory or marketing decisions is commonly referred to as demand elasticity.

   Most models that consider demand variation in response to item availability (i.e. inventory level) assume that the holding cost is constant for the entire inventory cycle. This paper presents an inventory model with a linear stock-level dependent demand rate and a quantity dependent holding cost. In this model, the holding cost is adecreasing step function of the time spent in storage.

   This structure is representative of many real-life situations in which distinctive unit holding cost occurs depends on the quantity kept in warehouse. This is particularly true in the storage of deteriorating and perishable items such as fruits in which quantity decreasing every time so the holding cost.

2. Problem definition and scope

   The main objective of this paper is to determine the optimum (i.e. minimum cost) inventory policy for an inventory system with inventory-level dependent demand rate and a quantity- dependent holding cost. Assuming the demand rate to be inventory-level dependent means the demand is higher for greater inventory levels. Assuming the holding cost per unit of the item per unit time to be quantity-dependent means the unit holding cost is lower for small number of items in warehouse. The model that will be developed for the inventory system is based on allowing unit holding cost values to vary with different quantity. Variable unit holding costs are considered in the model in determining the optimal inventory policy.

   The holding cost per unit is assumed to decrease only when the quantity reach to specified discrete values. In other words, the holding cost per unit per unit time is adecreasing step function of the quantity.

   Two types of holding cost step functions are considered: Retroactive decrease, and stepwisedecrease. In retroactive decrease, the unit holding cost rate of the first storage period is applied to all storage periods. In stepwisedecrease, the rate of each period, including the last period, is applied only to units stored in that particular period.

   1. Notation

      q (t) =The quantity on-hand at time t R= Constant demand rate

      N= Number of distinct time periods with different holding cost rates

      t =Time

      ti= End time of period i, where i= 1,2,3..n t0= 0, and tn =

      A= Ordering cost per order hi=Holding cost of the item in period i

      h (t) = Holding cost of an item at time t, h(t) = hi if

      T = Cycle time

      = Demand parameter indicating elasticity in relation to the inventory level

   2. Assumption and limitations

1. The demand rate R is linearly increasing function of the inventory level q.

2. The holding cost is varying as decreasing step function of the quantity in storage.

3. Replenishments are instantaneous.

4. Shortages are not allowed.

5. A single item is considered.

6. The demand rate R islinear function of the inventory level q which is expressed as

1. Inventory Model

   Our main objective is to minimize the TIC per unit time, which includes two components: The ordering cost, and the holding cost. Since one order is made per cycle,

   the ordering cost per unit time is simply and the total holding cost as follows.

   Since the demand rate is equal to the rate of inventory level decrease, we can describe inventory level by the following differential equation:

   Here we have

   Hence,

   Now the period T can be evaluated by putting

   So,

2. Retroactive holding cost decrease

   As stated earlier, the holding cost is assumed to be decreasingstep function of the quantity in storage, i.e. In this case uniform holding is used and the holding cost of the first storage period is applies retroactively to all subsequent periods. Thus the holding cost rate is applied to all periods i = 2, 3e. In this case, the TIC per unit time can be expressed as

   Substituting the value of from (2.3.3) we get;

   Now substituting value of T from (2.3.4) we get;

   On solving the above equation for Q we will get value of required order quantity Q.

3. Stepwise holding cost decrease

   The holding cost is now assumed to be andecreasingstep function of the quantity in storage. According to this function the holding cost rates applied to period 1, rate applied to period 2 and so on. Now the total inventory cost obtained as follows.

   Substituting the value of from (2.3.3), we obtain:

   Substituting the value of from (2.3.4), we obtain:

   To find the quantity Q, we set the derivative of TIC with respect to Q equal to zero.

4. Conclusions and suggestions

   A model has been presented of an inventory system with stock-dependent demand, in which the holding cost is a decreasingstep function of the quantity in storage. Two types of holding cost variation in terms of storage time have been considered: retroactive increase, and stepwise holding cost decrease based on the formulas developed, it can be concluded that both the optimal order quantity and the cycle time decrease when the holding cost increases.The model presented in this study provides a basis for several possible extensions. For future research, this model can be extended to accommodate plannedshortages, variable ordering costs, and non-instantaneous receipt of orders. The case of the decreasing holding cost considered in this paper applies rented storage facilities, where lower rent rates are normally obtained for longer-term leases.

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